Ideal Points Of Character Varieties, Algebraic Non-integral Representations, And Undetected Closed Essential Surfaces In 3-manifolds

Closed essential surfaces in a 3-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representation into PSL2(C), resolving a question of Schanuel and Zhang.